Performance of Pairs Trading on the S&P 500

Performance of Pairs Trading on the S&P 500: Distance and Copula-GARCH models Fernando A. Boeira Sabino da Silva∗, Flavio A. Ziegelmann†, and João F. Caldeira

§ ‡

Abstract We carry out a study to evaluate and compare the performance of the distance and copula strategies. Using data from the S&P500 shares from 1990 to 2015, we find that the Copula-GARCH approach yields significant higher returns and Sharpe ratios than the traditional distance approach when fewer restrictions are attached to trade under different weighting structures. Particularly, the Copula-GARCH and distance methods show a mean annualized excess return before costs on committed and fully invested capital as high as 6.33% and 5.29% and 17.53% and 9.27%, respectively. The Copula approach also presents more trading opportunities than the distance strategy. To test the statistical significance of the excess returns and Sharpe ratios we use the stationary bootstrap of Politis and Romano (1994) adopting the automatic block-length selection of Politis and White (2004). The Copula-GARCH strategy shows positive and significant alphas during the sample period after accounting for various risk-factors. Keywords: Copula; Pairs Trading; S&P 500; Stationary Bootstrap; Statistical Arbitrage. JEL Codes: C51, C58, G11.

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Introduction The performance of pairs trading strategies has been recently discussed in several studies with much

interest in empirical finance, since the strategy has potential to achieve profits through relatively low-risk positions. In addition, the strategy is claimed to be market neutral, which means that the investors are not exposed to market risk. The strategy was pioneered by Gerry Bamberger and later led by Nunzio Tartaglia’s quantitative group at Morgan Stanley in the 1980s. However, it became popular through the study carried out by Gatev, Goetzmann, and Rouwenhorst (2006), named distance method. Currently, there are three main strategies for pairs trading: distance, cointegration, and copula. In this paper, we will conduct an empirical investigation to offer some evidence of the behavior of the distance and copula strategies under different investment scenarios. We are also interested in verifying how the delay to start the positions and the trading costs affect the profitability of these strategies. The performance of the distance method has been measured thoroughly using different data sets and financial markets (Gatev et al., 2006; Perlin, 2009; Do and Faff, 2010, 2012; Broussard and Vaihekoski, 2012; Caldeira and Moura, 2013; Rad et al., 2015). In an efficient market, strategies based on meanreversion concepts should not generate consistent profits. However, Gatev, Goetzmann, and Rouwenhorst ∗ Department of Statistics, Federal University of Rio Grande do Sul, Porto Alegre, RS 91509-900, Brazil, e-mail: [email protected]; Corresponding author. † Department of Statistics, Federal University of Rio Grande do Sul, Porto Alegre, RS 91509-900, Brazil, e-mail: [email protected] ‡ Department of Economics, Federal University of Rio Grande do Sul, Porto Alegre, RS 90040-000, Brazil, e-mail: [email protected] § We thank Cristina Tessari for her extensive suggestions and for helping us obtaining the data we use.

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(2006) find that pairs trading generates consistent statistical arbitrage profits in the U.S. equity market during 1962-2002 using CRSP data, although the profitability declines over the period. They obtain a mean excess return above 11% a year during the reported period. The authors attribute the abnormal returns to a non-identified systematic risk factor. They support their view by showing that there is a high degree of correlation between the excess returns of non-overlapping top pairs even after accounting for risk factors from an augmented version of Fama and French (1993)’s three factors. Do and Faff (2010) extend their work by expanding the data sample and also find a declining trend - 33 basis points (bps) mean excess return per month for 2003-09 versus 124 basis points mean excess return per month for 1962-88. The authors note that there seems to have a higher risk of non-convergence in the final subperiod. Do and Faff (2012) show that the distance method is unprofitable after 2002 when trading costs are considered. Broussard and Vaihekoski (2012) test the profitability of pairs trading under different weighting structures and trade initiation conditions using data from the Finnish stock market. They also find that their proposed strategy is profitable even after initiating the positions one day after the signal. Rad, Low, and Faff (2015) evaluates distance, cointegration, and copula methods using a long-term and comprehensive data set. They also find that there is a decline in profitability using more sophisticated methods. The distance strategy (Gatev et al., 2006) uses the distance between normalized prices to capture the degree of mispricing between stocks. According to Xie, Liew, Wu, and Zou (2016), the distance method has a multivariate normal nature since it assumes a symmetric distribution of the spread between the normalized prices of the stocks within a pair and it uses a single distance measure, which can be seen as an alternative measurement of the linear association, to describe the relationship between two stocks. We know that if the series have joint normal distribution, then the linear correlation fully describes the dependence between the stocks. However, it is well known that the dependence between two securities are rarely jointly normal (Campbell et al., 1997; Cont, 2001; Ane and Kharoubi, 2003; McNeil et al., 2015). The main feature of joint distributions characterized by tail dependence is the presence of heavy and possibly asymmetric tails, thus the traditional hypothesis of (multivariate) Gaussianity is completely inadequate. Therefore, a single distance measure may fail to catch the dynamics of the spread between a pair of securities, and thus initiate and close the trades at non-optimal positions. Due to the complex dependence patterns of financial markets, a high-dimensional multivariate approach to tail dependence analysis is surely more insightful than assuming multivariate normal returns. Because of its flexibility, copulas are able to model better the empirically verified regularities normally attributed to multivariate financial returns: (1) asymmetric conditional variance with higher volatility for large negative returns and smaller volatility for positive returns (Hafner, 1998); (2) conditional skewness (Ait-Sahalia and Brandt, 2001; Chen et al., 2001; Patton, 2001); (3) Leptokurdicity (Tauchen, 2001; Andreou et al., 2001); and (4) nonlinear temporal dependence (Cont, 2001; Campbell et al., 1997). Thus, to address these issues, Liew and Wu. (2013) propose a pairs trading strategy based on bivariate copulas. However, they evaluate its performance using only three pre-selected pairs over a period of less than three years. Xie, Liew, Wu, and Zou (2016) employ a similar methodology over a ten-year period with 89 stocks. Both studies find that the performance of the copula strategy is superior to the distance strategy. Xie and Yuan (2013) set out the distance and cointegration approaches as special cases of copulas under certain regularity conditions. The authors also recommend further research on how to incorporate copulas into the pairs selection. It is suggested that there is a possibility of larger profits in terms of returns since copulas deals better with non-linear dependency structures. However, the approach can result in inferior performance due to issues as overfitting. Rad, Low, and Faff (2015) use a more comprehensive data set consisting of all the shares traded in the US market from 1962 to 2014. Meanwhile, they find an opposite result. Particularly, the distance, cointegration, and Copula-GARCH strategies show a mean monthly 2

excess return of 36, 33, and 5 bps after transaction costs and 88, 83, and 43 bps before transaction costs. In this paper, we employ a Copula-GARCH model, which consists in fitting, initially, the daily returns of the formation period using an ARMA(p,q)-GARCH(1,1) to model the marginals. For each pair, we test the following elliptical and Archimedean copula functions: Gaussian, t, Clayton, Frank, Gumbel, and one Archimedean mixture copula consisting of the best combination of Clayton, Frank, and Gumbel copulae. We compare both specifications with the distance strategy using two different thresholds: (1) 2σ as in Gatev, Goetzmann, and Rouwenhorst (2006), where σ stands for the historical standard deviation between two normalized price deviations, estimated during the formation period and (2) 0.75σ based on Vidyamurthy (2004). Following Gatev, Goetzmann, and Rouwenhorst (2006), we calculate returns using two weighting schemes: the return on committed capital and on fully invested capital. The former commits1 equal amounts of capital to each one of the pairs even if the pair has not been traded2 , whereas the latter divides all capital between the pairs that are open. To investigate the robustness of our results to bid-ask bounce (Jegadeesh (1990); Jegadeesh and Titman (1995); Conrad and Kaul (1989)) we follow Gatev, Goetzmann, and Rouwenhorst (2006) and use a “one-day rule”, i.e., we evaluate the performance of the strategies when positions are initiated on the day after the price divergence and closed on the day after the convergence. The authors reported considerably lower returns when they enforce this restriction. We also conduct the analysis using five subperiods to avoid the concern that the results are good only for the whole period: (1) July, 1990 to December, 1999; (2) January, 2000 to December, 2002; (3) January, 2003 to June 2007; (4) July, 2007 to June 2009; and (5) July, 2009 to December, 2015. The second subperiod corresponds to the bear market that spans the Dotcom crisis and the September 11th terrorist attack. The fourth subperiod corresponds to the subprime mortgage crisis. We compare the performance out-of-sample of the strategies using a variety of criteria, all computed using a rolling period procedure similar to that used by Gatev, Goetzmann, and Rouwenhorst (2006) with the exception that the time horizon of formation and trading periods are rolled forward by six months as in Broussard and Vaihekoski (2012). The main criteria we focus are: (1) mean excess return, (2) Sharpe and Sortino ratios, (3) average number of pairs opened per six-month period, (4) t-values for various risk factors, and (5) maximum drawdown between two consecutive days and between two days within a period of maximum six months. In order to find whether pairs trading profitability is associated to exposure to different systematic risk factors3 , we regress daily excess returns on different factors: (i) daily Fama and French (2015)’s five research factors

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and (ii) similarly to Gatev, Goetzmann, and Rouwenhorst (2006) we use another

version of daily Fama and French’s research factors including the daily Fama and French (1993)’s three factors plus momentum and short-term reversal. To test the statistical significance of the returns and Sharpe ratios we use the stationary bootstrap of Politis and Romano (1994) using the automatic block-length selection of Politis and White (2004) and 10,000 bootstrap resamples. To compute the bootstrap p-values we use the methodology proposed by 1 We

assume zero return for non-open pairs, although in practice one could earn returns on idle capital. committed capital is considered more realistic as it takes into account the opportunity cost of the capital that has been allocated for trading. 3 The single-factor capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965), as well as its consumption based version (Breeden (1979)), among other extensions, has been empirically tested and rejected by numerous studies, which show that the cross-sectional variation in expected equity returns cannot be explained by the market beta alone, providing evidence that investors demand compensation for not being able to diversify firm-specific characteristics. 4 Fama and French (2015) found evidences that the three factor model was not sufficient to explain a lot of the variation in average returns related to profitability and investment. 2 The

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Ledoit and Wolf (2008). We aim to compare the results on a statistical basis to mitigate potential data snooping problems. Our results indicate that the Copula strategy outperforms consistently the benchmark strategies for Top 5 and Top 20 pairs in the long term in terms of wealth accumulation and Sharpe Ratio when a fast trade can be executed. We also find that the intercept is statistically greater than zero for virtually all regressions without delay and/or before costs when considering the Copula-GARCH strategy, usually at 1% level, showing that our results are robust to Fama and French (1993) and Fama and French (2015)’s risk adjustment factors. Moreover, we also verify that the copula strategy identifies more trading opportunities than the distance strategy. In addition, the share of observations with negative excess returns is smaller for the copula than distance strategies. The principal contribution of this paper is to assess whether a more sophisticated approach than the distance method can take advantage of any market inefficiencies. This paper is one attempt to fill the gap in the literature through a proper analysis on this issue using a long-term and comprehensive data set. The main goal is to provide investors with information that allows for a more accurate selection of which strategy should be used in a particular setting and finally to understand any differences in performances. The remainder of the paper is organized as follows. Section 2 discusses the data, a general review of the distance and copula models, and the trading strategies we have used. Section 3 summarizes the empirical results of the analysis. Finally, Section 4 provides a brief conclusion. Additional results are reported in the Appendix.

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Data and Methodology Our data set consists of daily data of adjusted closing prices of all shares that belong to the S&P500

market index from July 2nd, 1990 to December 31st, 2015. We obtain the adjusted closing prices from Bloomberg and the returns on the Fama and French factors from French’s website5 . The data set sample period is made up of 6,426 days and includes a total of 1100 stocks over all periods. Only stocks that are listed during the formation period are included in the analysis, i.e., around 500 stocks in each trading period. Using data from the Center for Research in Security Prices (CRSP) from 1980 to 2006, French (2008) estimates that the cost of active investing, including total commissions, bid-ask spreads, and other cost investors pay for trading services, has dropped from 146 basis points in 1980 to 11 basis points in 2006. Considering the US stock live trades on the Nyse-Amex between August 1998 and September 2013 for a large institutional investor, Frazzini, Israel, and Moskowitz (2015) estimate that the average trading costs for market impact (MI) and implementation shortfall methodology (IS) are 8.81 and 9.13 basis points, respectively, while the median trading costs are 6.24 and 7.63 basis points, respectively. In this context, we assume trading costs on the order of 10 basis points in the following analysis.

2.1

Distance Framework

Our implementation of the distance strategy is similar to Gatev, Goetzmann, and Rouwenhorst (2006) and Broussard and Vaihekoski (2012). We calculate the spread between the normalized daily closing prices (known as distance) of all combinations of stocks pairs during the next 12 months, named formation 5 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/det_st_rev_factor_daily.html

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period, adjusting them by dividends, stock splits and other corporate actions. Specifically, the pairs are formed using data from January to December or from July to June. Prices are scaled to $1 at the beginning of each formation period and then evolve using the return series. We then select the top 5, 20, and 101-120 of those combinations that have the smallest sum of squared spreads, allowing re-selection of a specific pair, during the formation period. These pairs are then traded in the next six months (hereafter trading period). To implement the pairs trading procedure, we follow Gatev, Goetzmann, and Rouwenhorst (2006). Potential security pairs are sorted based on the sum of squared differences between their normalized prices during the formation period. In Gatev, Goetzmann, and Rouwenhorst (2006), when the spread diverges by two or more standard deviations (which is calculated in the formation period) from the mean, the stocks are assumed to be mispriced in terms of their relative value to each other and thus we open a short position in the outperforming stock and a long in the underperforming one. We will profit in the long run if these relative prices revert to their historical mean. According to Vidyamurthy (2004), with the assumption that the spread follows a Gaussian white noise process, the threshold that yields the highest profit is, approximately, 0.75 standard deviations 6 . The positions are closed once the normalized prices cross. The pair is then monitored for another divergence and thus a pair can complete multiple round-trip trades. Contrary to profit-making trades, trades that do not converge can result in a loss if they are still open at the end of the trading period when they are automatically closed. This results in fat left tails. Therefore, since the conditional variance is empirically higher for large negative returns and smaller for positive returns, it may be inappropriate to use constant trigger points because the volatility differs at different price levels. To calculate the daily percentage returns for a pair, we compute rpt = w1t rtL − w2t rtS ,

(1)

where L and S stands for long and short, respectively. Returns rpt can be interpreted as excess returns since in (1) the risk-free rate is canceled out when one calculates the long and short excess returns. The weights w1t and w2t are initially assumed to be one. After that, they change according to the changes in the value of the stocks, i.e., wit = wit−1 (1 + rit−1 ).

2.2

Copula Framework

The notion of copula was first introduced by Sklar (1959). Sklar’s theorem states that any multivariate joint distribution can be written in terms of their univariate marginal distribution function and the dependence structure between the variables. We state the Sklar’s Theorem below. Theorem 1. (Sklar’s Theorem) Let X1 , . . . , Xd be random variables with distribution functions F1 , . . . , Fd , respectively. Then, there exists an d-copula C such that, F (x1 , . . . , xd ) = C (F1 (x1 ) , . . . , Fd (xd )) ,

(2)

for all x = (x1 , . . . , xd ) ∈ Rd . Conversely, if C is an d-copula and F1 , . . . , Fd are distribution functions, then the function F defined by (2) is a d-dimensional distribution function with margins F1 , . . . , Fd . Fur6 If

normally distributed, 0.75σ and 2.0σ would happen 45% and 5% of the time, approximately.

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thermore, if the marginals F1 , . . . , Fd are all continuous, C is unique. Otherwise C is uniquely determined on Im F1 × . . . × Im Fd . Using the Sklar’s theorem we can derive an important relation between the marginal distributions and a copula. Let f be a joint density function (of the d-dimensional distribution function F ) and f1 , . . . , fd univariate density functions of the margins F1 , . . . , Fd . Assuming that F (·) and C(·) are differentiable, by (2) we have ∂ d F (x1 , . . . , xd ) ∂x1 . . . ∂xd

≡ f (x1 , . . . , xd ) = =

c (u1 , . . . , ud )

d Y

∂ d C (F1 (x1 ) , . . . , Fd (xd )) ∂x1 . . . ∂xd

(3)

fi (xi ) .

(4)

i=1

Thus, copulas are functions that connect a multivariate distribution function and their marginal distributions. Thereafter, copulas conveniently separate marginals from dependence component and accommodate various forms of dependence through suitable choice of the copula“correlation matrix”. These carry on all relevant information about the dependence structure between random variables and allow a greater flexibility in modeling multivariate distributions and their margins. The methodology allows one to derive joint distributions from marginals, even when these are not normally distributed. In fact, copulas allow the marginal distributions to be modeled independently from each other, and no assumption on the joint behavior of the marginals is required, which provides a great deal of flexibility in modeling joint distributions. The choice of the copula function is also not dependent on the marginal distributions. Thus, by using copulas, the linearity restriction that applies to the dependence structure of multivariate random variables in a traditional dependence setting is relaxed. Therefore, depending on the chosen copulas, different dependence structures can be modeled to allow for any asymmetries7 . A further important property of copulas concerns the partial derivatives of a copula with respect to its variables. Let now H be a bivariate function with marginal distribution functions F and G. According 2

to Sklar (1959) then there exists a copula C : [0, 1] → [0, 1] such that H(x, y) = C(F (x), G(y)) for all x, y ∈ R2 . If F and G are continuous, then C is unique; otherwise, C is uniquely determined in Im F × Im G. Conversely, if C is a copula and F and G are distribution functions, then the function H is a joint distribution function with marginals F and G and we can write C(u, v) = H(F −1 (u), G−1 (v)),

(5)

where u = F (x) ⇒ x = F −1 (u), v = G(y)) ⇒ y = G−1 (v), and F −1 and G−1 are the quasi-inverses of F and G, respectively. For any copula C,

∂C(u,v) ∂u

and

∂C(u,v) ∂u

exist almost everywhere. The proposition

below states that the partial derivatives of a copula function corresponds to the conditional probabilities of the random variables (see Cherubini et al., 2004; Nelsen, 2006).

7 Copulas measures lower and upper tail dependencies and nonlinear and linear relationships in a rich set for describing dependencies between pairs. Copula is also invariant under strictly monotonic transformations (Cherubini et al., 2004; Nelsen, 2006) and hence the same copula is obtained if we use price or return series, for example.

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Proposition 1. Let U and V be two random variables with distribution U (0, 1). Then, ∂C (u, v) = P (U ≤ u |V = v ) , ∂v ∂C (u, v) (ii) = P (V ≤ v |U = u ) , ∂u (i)

where ∂C (u, v) = P (U ≤ u |V = v ) = lim P (U ≤ u |v ≤ V ≤ v + h ) , h→0 ∂v

(6)

∂C (u, v) = P (V ≤ v |U = u ) = lim P (V ≤ v |u ≤ U ≤ u + h ) . h→0 ∂u

(7)

and

By using the fact that the partial derivative of the copula function gives the conditional distribution function, Xie, Liew, Wu, and Zou (2016) define a measure to denote the degree of mispricing: Definition 1. (Mispricing Index) If RtX and RtY represent the random variables of the daily returns of stocks X and Y on time t, and the realizations of those returns on time t are rtX and rtY , we have t M IX|Y =P (RtX < rtX | RtY = rtY )

and M IYt |X

=P (RtY < rtY | RtX = rtX ),

where M IX|Y and M IY |X are named the mispricing indexes. t Therefore, the conditional probabilities M IX|Y and M IYt |X indicate whether the return of X is con-

sidered high or low at time t, given the information on the return of Y on the time t and the historical t relation between the two stocks’ returns, and vice-versa. For example, if the value of M IX|Y is equal

to 0.5, rtX is neither too high nor too low given rtY and their historical relation. In other words, the historical data indicate that, on average, there is an equal number of observations of the return of X being larger or smaller than rtX if the return of stock Y is equal to rtY . In this case, we can say that stock X is fairly priced relative to stock Y on day t. Considering the potential problems when using the spread as a measure, this measure should provide a comparable value that reflects the level of mispricing and it also must be consistent regardless of the price level or period. Given current realizations rtX and rtY , if FX and FY are the marginal distribution functions of RtX and RtY and C is the copula connecting FX and FY , we define u = FX rtX and v = FY rtY , and have t M IX|Y =

∂C(u, v) ∂v (8)

and M IYt |X =

∂C(u, v) . ∂u

In our copula approach, we find that 0.75 is a good combination for the mispricing indexes during our backtesting analysis8 . This would be similar to the 0.75σ trigger point if data is normally distributed. We fit nominated pairs to copulas in two steps. First, for each pair, we fit an appropriate ARMA8 We

search over a grid from 0.55 to 0.95 with a step of size 0.05.

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GARCH(1,1) model9 to each univariate time series (daily returns of the formation period) by obtaining the estimates µ bi and σ bi of the conditional mean and variance of these processes, respectively. Moreover, using the estimated parametric models, we construct the standardized residuals vectors given, for each i = 1, . . . , t, by εbi =

xi − µ bi . σ bi

The standardized residuals vectors are then converted to the pseudo-observations zi =

(9) n n+1 Fi

(b εi ),

where Fi is estimated by using their empirical distribution function. In the second step, with the estimated parameters from the previous step, we nominate the copula that best fits the uniform marginals and estimate its parameter(s). Copulas that are tested in this step are Gaussian, t, Clayton, Frank, and Gumbel for the Copula-GARCH model and Gaussian, t and one Archimedean mixture copula consisting of the optimal linear combination of Clayton, Frank and Gumbel copulas for the Mixture Copula-GARCH approach10 . Specifically, a mixture of the Clayton, Frank, and Gumbel copulas can be written as CθCF G (u, v) = π1 CαC (u, v) + π2 CβF (u, v) + (1 − π1 − π2 ) CδG (u, v) ,

(10)

0

where θ = (α, β, δ) are the Clayton, Frank, and Gumbel copula (dependence) parameters, respectively, and π1 , π2 ∈ [0, 1]. The estimates are obtained by the minimization of the negative log-likelihood consisting of the weighted densities of the Clayton, Frank, and Gumbel copulas. Xie, Liew, Wu, and Zou (2016) propose the following steps to obtain M IX|Y and M IY |X using copulas: (1) Calculate daily returns for each stock during the training period and estimate the marginal distributions of returns separately; (2) After obtaining the estimated marginal distributions, we estimate the copula C that best fits to the used data to connect the joint distributions with the marginals FX Y and FY ; (3) During trading period, the (adjusted) closing prices pX t and pt are used to calculate daily

returns rtX and rtY . Therefore M IX|Y and M IY |X can be calculated for each day in the trading period using the copula and estimated parameters given in (2); (4) Build long and short positions of Y and X on the days that M IX|Y > ∆1 and M IY |X < ∆2 if there is no positions in X or Y . Conversely, build positions long/short of X and Y on the day that M IX|Y < ∆2 and M IY |X > ∆1 if there is no positions in X or Y . All positions are closed if M IX|Y reaches ∆3 or M IY |X reaches ∆4 , where ∆1 , ∆2 , ∆3 and ∆4 are predetermined thresholds or are automatically closed out on the last day of the trading period if they do not reach the thresholds. Here we use ∆1 = 0.75, ∆2 = 0.25 and ∆3 = ∆4 = 0.5.

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Empirical Results An empirical study is carried out to evaluate and compare the performance of the distance and

copula strategies. Such study is necessarily restrictive because there are many possibilities regarding the selection of the triggering points, the trading costs, the number of pairs, the type of volatility model, the probability density functions, the type of squared error criterion function, among other factors. 9 We

look for the best ARMA(p,q) model up to order (1,1). three Archimedean copulas contain different tail dependence characteristics. Clayton and Gumbel are nonsymmetric copulas that describe more accurately lower and upper tail dependence, respectively. The Frank copula is the only bivariate reflection symmetric Archimedean family but it has different properties when compared to the bivariate Gaussian and bivariate t copulas. Hence, by using a mixture copula we cover a wider range of possible dependencies within a single model. 10 These

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3.1

Profitability of the Strategies

Tables 1 to 6 report annualized excess returns, annualized Sharpe and Sortino ratios, Newey and West (1987) adjusted t-statistics, share of negative observations, the maximum drawdown in terms of maximum percentage drop between two consecutive days (MDD1) and between two days within a period of maximum six months (MDD2) for each of the three strategies from 1990/2-2015, for the Top 5, Top 20, and Top 101-120 pairs. Tables 1 to 3 summarize results when positions are initiated and closed in the same day the pair diverges, whereas Tables 4 to 6 when we delay trades by one day. Furthermore, Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. By analyzing Tables 1 to 2, it is possible to observe a series of important facts. First, notice that the copula-based pairs strategy outperforms the distance strategies when considering the Sharpe and Sortino risk-return ratios for Top 5 and Top 20 pairs, particularly before costs. The copula strategy yields a Sharpe ratio on committed capital above 1 for Top 20 pairs (1.72 and 1.03 before and after costs, respectively), indicating that the returns on committed capital are greater than the risk taken. The Sortino ratio confirms that the copula method offers better risk-adjusted returns. The statistics also indicate that the copula model delivers the highest t-statistics (all statistically and economically significant at 1%) and a lower probability of a negative trade, especially for Top 5 pairs, where the share of days with negative returns (38%-39%) is consistently less than the market performance (47.45% of negative returns over the period).

Table 1: Strategy

Excess returns of pairs trading strategies on portfolios of Top 5 pairs without delay. Mean Return

Sharpe ratio

Sortino ratio

t-stat

% of negative trades

MDD1

MDD2

46.83 48.71 39.14

-6.12 -5.81 -6.10

-18.85 -16.65 -16.69

46.78 48.66 38.05

-6.12 -5.80 -6.09

-18.86 -16.54 -15.49

46.83 48.70 39.11

-7.87 -7.62 -10.40

-28.41 -30.52 -32.09

46.78 48.66 38.05

-7.87 -7.62 -10.40

-28.31 -30.40 -30.10

Section 1: Return on Committed Capital Panel A - After Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

3.83 4.59 3.53

0.46 0.50 0.57

2.65∗∗∗ 2.88∗∗∗ 3.15∗∗∗

0.85 0.94 0.94

Panel B - Before Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

4.14 5.19 6.33

0.50 0.56 0.99

2.83∗∗∗ 3.21∗∗∗ 5.38∗∗∗

0.92 1.05 1.64

Section 2: Return on Fully Invested Capital Panel A - After Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

7.41 4.59 9.06

0.49 0.36 0.53

2.96∗∗∗ 2.19∗∗ 3.16∗∗∗

1.00 0.72 1.00

Panel B - Before Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

7.91 5.39 16.83

0.53 0.42 0.94

3.13∗∗∗ 2.49∗∗ 5.28∗∗∗

1.05 0.83 1.70

Note: Summary statistics of the annualized excess returns, annualized Sharpe and Sortino ratios on portfolios of top 5 pairs between July 1991 and December 2015 (6,173 observations). The positions are initiated in the same day the pair diverges. The t-statistics are computed using Newey-West standard errors with a six-lag correction. The columns labeled MDD1 and MDD2 compute the largest drawdown in terms of maximum percentage drop between two consecutive days and between two days within a period of maximum six months, respectively. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

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Table 2: Strategy

Excess returns of pairs trading strategies on portfolios of Top 20 pairs without delay. Mean Return

Sharpe ratio

Sortino ratio

t-stat


MDD1

MDD2

47.85 47.53 47.46

-4.27 -4.18 -4.07

-11.38 -13.76 -8.83

47.68 47.30 45.07

-4.28 -4.18 -4.06

-11.31 -13.67 -8.19

47.85 47.53 47.45

-5.51 -4.85 -12.52

-24.74 -22.53 -25.45

47.68 47.30 45.07

-5.50 -4.86 -12.52

-24.63 -22.38 -22.93

Section 1: Return on Committed Capital Panel A: After Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

2.85 3.98 3.66

0.57 0.72 1.03

2.94∗∗∗ 3.77∗∗∗ 5.18∗∗∗

1.00 1.27 1.77

Panel B: Before Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

3.13 4.54 6.29

0.63 0.81 1.72

3.21∗∗∗ 4.25∗∗∗ 8.63∗∗∗

1.09 1.44 3.02

Section 2: Return on Fully Invested Capital Panel A: After Transaction Costs Distance (2.0σ) Distance (0.75σ) Copula-GARCH

5.49 4.82 8.41

0.59 0.62 0.69

3.15∗∗∗ 3.36∗∗∗ 3.82∗∗∗

1.05 1.11 1.22


5.97 5.58 17.53

0.64 0.72 1.36

3.39∗∗∗ 3.84∗∗∗ 7.25∗∗∗

1.14 1.27 2.37

Note: Summary statistics of the annualized excess returns, annualized Sharpe and Sortino ratios on portfolios of top 20 pairs between July 1991 and December 2015 (6,173 observations). The positions are initiated in the same day the pair diverges. The t-statistics are computed using Newey-West standard errors with a six-lag correction. The columns labeled MDD1 and MDD2 compute the largest drawdown in terms of maximum percentage drop between two consecutive days and between two days within a period of maximum six months, respectively. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

Overall, the summary statistics show that copula method is a less risky strategy for Top 5 and Top 20 pairs, except when considering the drawdown measures. In this case, we obtain mixed results. We can note that the copula strategy is superior or very comparable to the distance strategies on committed capital, particularly when considering a period of six months (MDD2) and is usually the worst on fully invested capital, especially for the drop between two consecutive days (MDD1). The copula-based method also shows the highest average excess returns on fully invested capital (up to 17.53%) and before costs on committed capital. Although the outcomes are not the best on committed capital after costs, the strategy still has a competitive advantage since it delivers the highest risk-adjusted statistics among the strategies after considering the costs. Table 3 displays the results for Top 101-120 pairs, i.e., 20 pairs below the Top 100 pairs. The distance methods appear to have a better out-of-sample performance than in the Top 5 and Top 20 pairs, yielding higher values for most of the performance statistics computed. Compared to the copula strategy, the performance of the distance approaches is consistently better after costs, except for MDD2 on committed capital.

10

Table 3:

Excess returns of pairs trading strategies on portfolios of Top 101-120 pairs without delay.

Strategy

Mean Return

Sharpe ratio

Sortino ratio

t-stat


MDD1

MDD2

47.87 48.10 48.05

-3.32 -3.18 -4.10

-9.43 -8.94 -8.79

47.72 47.92 46.02

-3.31 -3.16 -4.09

-9.38 -8.91 -8.13

47.85 48.10 48.05

-6.64 -6.41 -9.06

-22.87 -18.40 -26.43

47.72 47.92 46.02

-6.63 -6.40 -9.02

-22.74 -18.13 -24.49


5.01 4.78 2.18

0.88 0.76 0.53

4.67∗∗∗ 4.09∗∗∗ 2.81∗∗∗

1.62 1.38 0.89


5.28 5.29 4.43

0.93 0.84 1.06

4.89∗∗∗ 4.48∗∗∗ 5.50∗∗∗

1.70 1.52 1.77


8.79 5.85 1.49

0.77 0.63 0.11

4.28∗∗∗ 3.48∗∗∗ 0.91

1.44 1.17 0.28


9.27 6.57 9.31

0.81 0.71 0.64

4.48∗∗∗ 3.86∗∗∗ 3.66∗∗∗

1.51 1.30 1.13

Note: Summary statistics of the annualized excess returns, annualized Sharpe and Sortino ratios on portfolios of top 101-120 pairs between July 1991 and December 2015 (6,173 observations). The positions are initiated in the same day the pair diverges. The t-statistics are computed using Newey-West standard errors with a six-lag correction. The columns labeled MDD1 and MDD2 compute the largest drawdown in terms of maximum percentage drop between two consecutive days and between two days within a period of maximum six months, respectively. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

Overall, we can notice that the copula strategy suffer considerably more when transaction costs are taking into account than the distance strategies. The excess returns of copula method are reduced by more than 40% after costs, while the distance strategies profits are reduced by no more than 15%.

3.2

Speed of execution

Up to now we have assumed that a trade can be started as soon as the trigger points are reached. However, one may face situations where the price of a stock bounces very quickly back and forth affecting how fast a trade can be executed. To minimize the effect of the bid-ask bounce on trading we report the effects of trading with one day delay. By analyzing Tables 4 to 5, we can notice that the copula strategy is, likewise after transaction costs, more sensitive to the speed of execution than the distance strategies for Top 5 and Top 20 pairs. The average excess returns on committed capital for copula and distance methods drop by about 3%-4% and less than 2% per year, respectively. This shows that a significant portion of the returns for the copula approaches occur in the first day of trading and thus may be due to bid-ask bounce. However, as highlighted by Gatev, Goetzmann, and Rouwenhorst (2006), it is difficult to quantify which portion of this reduction is due to bid-ask bounce and which portion is derived from the true mean reversion in prices because of fast market adjustment.

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Table 4: period.

Excess returns of pairs trading strategies on portfolios of Top 5 pairs with a one-day waiting

Strategy

Mean Return

Sharpe ratio

Sortino ratio

t-stat


MDD1

MDD2

46.70 48.65 45.28

-5.56 -5.29 -5.87

-18.91 -18.55 -15.54

46.64 48.49 39.61

-5.56 -5.28 -5.81

-18.91 -18.45 -14.83

46.67 48.83 44.94

-7.15 -7.32 -11.28

-27.31 -26.56 -35.45

46.61 48.68 39.83

-7.15 -7.32 -11.22

-27.21 -26.42 -34.01


2.26 3.38 -0.42

0.28 0.37 -0.07

1.66∗ 2.20∗∗ -0.19

0.54 0.73 -0.06


2.55 3.97 2.28

0.32 0.44 0.36

1.85∗ 2.53∗∗ 2.03∗∗

0.60 0.84 0.62


4.51 4.37 0.47

0.31 0.28 0.03

2.01∗∗ 1.83∗ 0.56

0.66 0.70 0.18


5.00 5.09 4.82

0.34 0.32 0.28

2.17∗∗ 2.06∗∗ 1.84∗

0.71 0.79 0.59

Note: Summary statistics of the annualized excess returns, annualized Sharpe and Sortino ratios on portfolios of top 5 pairs between July 1991 and December 2015 (6,173 observations). We assume a one day waiting period after the pair diverges. The t-statistics are computed using Newey-West standard errors with a six-lag correction. The columns labeled MDD1 and MDD2 compute the largest drawdown in terms of maximum percentage drop between two consecutive days and between two days within a period of maximum six months, respectively. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

Notice that the profits generated by the copula method are not statistically significant, after considering transaction costs and one-day waiting period restriction on execution. Meanwhile, the trading profits from distance approaches survive when implementing both constraints at least at 10%. Moreover, the distance methods yield higher returns even before costs for Top 5 pairs, although the results are comparable, particularly when considering risk-adjusted statistics on committed capital. For Top 20 pairs the copula model appears to yield competitive results after costs, when evaluating by excess returns, Sharpe and Sortino ratios. Table 6 displays the out-of-sample performance statistics with "one-day rule" for Top 101-120 pairs. The results are quite divergent from the other top pairs, since the copula model appears to be relatively less sensitive when a fast trade is not executed. Unintuitively, the method produces even higher profits after costs on fully invested capital, although still not significant. Compared to the distance methods, the copula-based pairs strategy delivers, before costs, the best risk-adjusted performance on committed capital, including drawdown measures and higher trading profits on fully invested capital. Little is known about the statistical properties of these estimators in practice, and therefore these out-of-sample results may be due to the random nature of the pairs.

12

Table 5: period.

Excess returns of pairs trading strategies on portfolios of Top 20 pairs with a one-day waiting

Strategy

Mean Return

Sharpe ratio

Sortino ratio

t-stat


MDD1

MDD2

48.02 48.27 50.32

-3.82 -3.73 -4.02

-9.86 -11.89 -11.52

47.85 47.97 47.40

-3.82 -3.73 -4.01

-9.75 -11.80 -10.36

47.61 48.27 49.26

-5.44 -6.31 -11.45

-19.48 -19.31 -30.96

47.48 47.95 47.38

-5.44 -6.32 -11.44

-19.37 -19.14 -29.16


1.84 2.63 0.43

0.38 0.48 0.12

2.04∗∗ 2.63∗∗∗ 0.69

0.67 0.87 0.23

Panel B: Before Transaction Costs Strategy Distance (2.0σ) Distance (0.75σ) Copula-GARCH

2.11 3.18 2.97

0.44 0.58 0.83

2.32∗∗ 3.14∗∗∗ 4.18∗∗∗

0.77 1.04 1.41


4.06 2.60 1.49

0.45 0.33 0.11

2.48∗∗ 1.89∗ 0.91

0.82 0.62 0.29


4.54 3.30 6.71

0.50 0.42 0.48

2.74∗∗∗ 2.33∗∗ 2.84∗∗∗

0.91 0.77 0.91

Note: Summary statistics of the annualized excess returns, annualized Sharpe and Sortino ratios on portfolios of top 20 pairs between July 1991 and December 2015 (6,173 observations). We assume a one day waiting period after the pair diverges. The t-statistics are computed using Newey-West standard errors with a six-lag correction. The columns labeled MDD1 and MDD2 compute the largest drawdown in terms of maximum percentage drop between two consecutive days and between two days within a period of maximum six months, respectively. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

Figures 1 to 4 show cumulative excess returns through the full data set for every strategy. Figure 1 shows the returns with no delay and before costs for Top 5 (top), Top 20 (center) and Top 101-120 pairs (bottom) on committed (left) and fully invested capital (right) whereas Figure 2 displays the returns with no delay and after costs, Figure 3 with “one-day rule” after the signal and before costs, and Figure 4 with “one-day rule” after the signal and after costs. The patterns found in the figures bolster the average excess returns and t-statistics displayed in Tables 1 to 6. Overall, we can observe that the copula strategy has a relatively stronger performance when no restrictions are imposed on trade, i.e., before costs and when trades can be executed quickly after the signal, in particular on fully invested capital. When only one of the constraints is attached to trade the performance measures show that the copula method still delivers a relative good out-performance over the whole data set. However, the distance strategies are clearly less affected by the restrictions for Top 5 and Top 20 pairs.

13

Table 6: Excess returns of pairs trading strategies on portfolios of Top 101-120 pairs with a one-day waiting period. Strategy

Mean Return

Sharpe ratio

Sortino ratio

t-stat


MDD1

MDD2

48.47 48.29 47.84

-3.28 -3.05 -2.81

-7.37 -7.22 -6.61

48.29 48.06 45.47

-3.24 -3.00 -2.78

-7.31 -7.19 -5.87

48.34 48.40 47.72

-4.92 -3.65 -9.20

-17.91 -14.55 -22.06

48.15 48.21 45.93

-4.91 -3.65 -9.17

-17.77 -14.40 -20.33


3.31 3.46 1.45

0.59 0.56 0.37

3.24∗∗∗ 3.09∗∗∗ 1.98∗∗

1.08 1.02 0.63

Panel B: Before Transaction Costs Strategy Distance (2.0σ) Distance (0.75σ) Copula-GARCH

3.57 3.96 3.70

0.64 0.63 0.93

3.47∗∗∗ 3.48∗∗∗ 4.80∗∗∗

1.16 1.16 1.56


5.24 4.66 2.89

0.48 0.50 0.19

2.83∗∗∗ 2.91∗∗∗ 1.35

0.93 0.94 0.42


5.70 5.34 8.00

0.52 0.57 0.50

3.04∗∗∗ 3.28∗∗∗ 2.96∗∗∗

1.01 1.07 0.93

Note: Summary statistics of the annualized excess returns, annualized Sharpe and Sortino ratios on portfolios of top 101-120 pairs between July 1991 and December 2015 (6,173 observations). We assume a one day waiting period after the pair diverges. The t-statistics are computed using Newey-West standard errors with a six-lag correction. The columns labeled MDD1 and MDD2 compute the largest drawdown in terms of maximum percentage drop between two consecutive days and between two days within a period of maximum six months, respectively. ∗ ∗∗ ∗∗∗ , , significant at 1%, 5% and 10% levels, respectively.

It should be noted that the copula strategy displays, from Figures 1 to 3, a very poor performance from 1998-2000 for Top 5 pairs. However, it achieves a very favorable out-of-sample performance relative to the benchmark approaches after the subprime mortgage crisis.

14

(a) Top 5 pairs, Committed Capital, no waiting, before costs

(b) Top 5 pairs, Fully Invested, no waiting, before costs 50

CC Copula−GARCH CC Distance (0.75σ) CC Distance (2.0σ)

4

Cumulative Return

Cumulative Return

5

3 2 1 0 91

93

95

97

99

01

03 05 Year

07

09

11

13

30 20 10 0 91

16

(c) Top 20 pairs, Committed Capital, no waiting, before costs

Cumulative Return

Cumulative Return

3 2 1

93

95

97

99

01

03 05 Year

07

09

11

13

99

01

03 05 Year

07

09

11

13

16


3 2 1

93

95

97

99

01

03 05 Year

07

09

11

13

16

40 30 20 10 93

95

97

99

01

03 05 Year

07

09

11

13

16

(f) Top 101−120 pairs, Fully Invested, no waiting, before costs 12 Cumulative Return

Cumulative Return

97

FI Copula−GARCH FI Distance (0.75σ) FI Distance (2.0σ)

50

0 91

16

(e) Top 101−120 pairs, Committed Capital, no waiting, before costs 4

Figure 1: delay

95

60 CC Copula−GARCH CC Distance (0.75σ) CC Distance (2.0σ)

4

0 91

93

(d) Top 20 pairs, Fully Invested, no waiting, before costs

5

0 91


40


10 8 6 4 2 0 91

93

95

97

99

01

03 05 Year

07

09

11

13

16

Cumulative excess returns of pairs trading strategies before costs and without

This figure shows how an investment of $1 evolves from July 1991 to December 2015 for every strategy.

15

(a) Top 5 pairs, Committed Capital, no waiting, after costs

(a) Top 5 pairs, Fully Invested, no waiting, after costs 10


3 2 1 0 91

93

95

97

99

01

03 05 Year

07

09

11

13


8

Cumulative Return

Cumulative Return

4

6 4 2 0 91

16

(a) Top 20 pairs, Committed Capital, no waiting, after costs

Cumulative Return

Cumulative Return

2 1.5 1

93

95

97

99

01

03 05 Year

07

09

11

13

01

03 05 Year

07

09

11

13

16


3 2 1

93

95

97

99

01

03 05 Year

07

09

11

13

16

6 4 2

93

95

97

99

01

03 05 Year

07

09

11

13

16

(a) Top 101−120 pairs, Fully Invested, no waiting, after costs 12 Cumulative Return

Cumulative Return

99


0 91

16

(a) Top 101−120 pairs, Committed Capital, no waiting, after costs 4

Figure 2: delay

97

8 CC Copula−GARCH CC Distance (0.75σ) CC Distance (2.0σ)

2.5

0 91

95

(a) Top 20 pairs, Fully Invested, no waiting, after costs

3

0.5 91

93


10 8 6 4 2 0 91

93

95

97

99

01

03 05 Year

07

09

11

13

16

Cumulative excess returns of pairs trading strategies after costs and without

This figure shows how an investment of $1 evolves from July 1991 to December 2015 for every strategy.

16


3 2.5 2 1.5 1 0.5 91

93

95

97

99

(a) Top 5 pairs, Fully Invested, wait one day, before costs 5 Cumulative Return

Cumulative Return

(a) Top 5 pairs, Committed Capital, wait one day, before costs 3.5

01

03 05 Year

07

09

11

13

1.5 1 0.5 91

93

95

97

99

01

03 05 Year

07

09

11

13

1.5 1 0.5 91

93

95

97

99

01

03 05 Year

07

09

11

13

16

95

97

99

01

03 05 Year

07

09

11

13

16


4 3 2 1 93

95

97

99

01

03 05 Year

07

09

11

13

16

(a) Top 101−120 pairs, Fully Invested, wait one day, before costs 8 Cumulative Return

Cumulative Return

2

93

5

0 91

16


1

(a) Top 20 pairs, Fully Invested, wait one day, before costs

(a) Top 101−120 pairs, Committed Capital, wait one day, before costs 3 2.5

2

6 Cumulative Return

Cumulative Return

(a) Top 20 pairs, Committed Capital, wait one day, before costs 2.5 2

3

0 91

16



4


6 4 2 0 91

93

95

97

99

01

03 05 Year

07

09

11

13

16

Figure 3: Cumulative excess returns of pairs trading strategies before costs and with oneday waiting period This figure shows how an investment of $1 evolves from July 1991 to December 2015 for every strategy.

17

(a) Top 5 pairs, Committed Capital, wait one day, after costs

(a) Top 5 pairs, Fully Invested, wait one day, after costs 5


2.5

Cumulative Return

Cumulative Return

3

2 1.5 1 0.5 91

93

95

97

99

01

03 05 Year

07

09

11

13

3 2 1 0 91

16

(a) Top 20 pairs, Committed Capital, wait one day, after costs

Cumulative Return

Cumulative Return

97

99

01

03 05 Year

07

09

11

13

16

1.5 1

93

95

97

99

01

03 05 Year

07

09

11

13

16

2.5 2 1.5 1 0.5 91

2 1.5 1

93

95

97

99

01

03 05 Year

07

09

11

13

16

93

95

97

99

01

03 05 Year

07

09

11

13

16

(a) Top 101−120 pairs, Fully Invested, wait one day, after costs 5 Cumulative Return


2.5


3

(a) Top 101−120 pairs, Committed Capital, wait one day, after costs 3 Cumulative Return

95

3.5 CC Copula−GARCH CC Distance (0.75σ) CC Distance (2.0σ)

2

0.5 91

93

(a) Top 20 pairs, Fully Invested, wait one day, after costs

2.5

0.5 91


4


4 3 2 1 0 91

93

95

97

99

01

03 05 Year

07

09

11

13

16

Figure 4: Cumulative excess returns of pairs trading strategies after costs and with oneday waiting period This figure shows how an investment of $1 evolves from July 1991 to December 2015 for every strategy.

3.3

Trading statistics

Table 7 reports trading statistics. Panel A, B and C report results for Top 5, Top 20, and Top 101120 pairs, respectively. The average price deviation trigger for opening pairs is listed in the first row of each panel. For any panel, we can observe that for the 0.75σ trigger the positions are initiated before, relatively. The positions are initiated when prices have diverged by 3.09%, 3.53%, and 5.02% for Top 5, Top 20, and Top 101-120, respectively. Similar to Gatev, Goetzmann, and Rouwenhorst (2006), the trigger spread increases with the number of pairs for all approaches.

18

Table 7:

Trading statistics.

Strategy

Distance (2.0σ)

Distance (0.75σ)

CopulaGARCH

Panel A: Top 5 Average price deviation trigger for opening pairs Total number of pairs opened Average number of pairs traded per six-month period Average number of round-trip trades per pair Standard Deviation Average time pairs are open in days Standard Deviation Median time pairs are open in days

0.0622 346 7.0612 1.4122 1.0389 50.8382 38.7356 39.5

0.0309 674 13.7551 2.7510 2.2560 33.5208 37.3741 16

0.0697 3288 67.1020 13.4204 7.7886 2.9793 4.3142 2

Panel B: Top20 Average price deviation trigger for opening pairs Total number of pairs opened Average number of pairs traded per six-month period Average number of round-trip trades per pair Standard Deviation Average time pairs are open in days Standard Deviation Median time pairs are open in days

0.0692 1302 26.5714 1.3286 0.9945 52.7366 40.1988 42

0.0353 2502 51.0612 2.5531 2.2627 35.3373 38.9524 17

0.0765 12242 249.8367 12.4918 7.8108 3.0734 4.5812 2

Panel C: Top 101-120 Average price deviation trigger for opening pairs Total number of pairs opened Average number of pairs traded per six-month period Average number of round-trip trades per pair Standard Deviation Average time pairs are open in days Standard Deviation Median time pairs are open in days

0.0969 1194 24.3673 1.2184 1.0234 54.7111 40.2876 45

0.0502 2210 45.1020 2.2551 2.0475 38.4715 40.2570 20

0.1046 10783 220.0612 11.0031 7.3947 3.3447 4.5950 2

Note: Trading statistics for portfolio of top 5, 20 and 101-120 pairs between July 1991 and December 2015 (49 periods). Pairs are formed over a 12-month period according to a minimum-distance criterion and then traded over the subsequent 6-month period. Average price deviation trigger for opening a pair is calculated as the price difference divided by the average of the prices.

The average number of pairs traded per six-month period is about 5 times more for copula approach than for the 0.75σ trigger, and more than 9 times the 2.0σ trigger over the 49 trading periods11 . Each pair is held open, in average, by approximately 3 trading days, which indicates that it is a short-term strategy when using the copula rules. Meanwhile, the average holding period for the 0.75 and 2.0 standard deviations approaches are about 1.6-1.8 and 2.5 trading months, respectively. This indicates that the strategy is a medium-term investment when using the distance approaches.

3.4

Regression on Fama-French asset pricing factors

To evaluate whether pairs trading profitability is a compensation for risk, we regress daily excess returns onto different risk factors: (i) daily Fama and French (2015)’s five research factors: the excess return on a broad market portfolio, (Rm − Rf ), the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks (SM B, small minus big), the difference between the 11 Clearly, one of the reasons for copula-based pairs strategy being more affected by trading costs is due to a much greater number of trading opportunities. Implementing a stop-loss strategy may lead to higher returns and reduce the standard deviation of returns.

19

return on a portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks (HM L, high minus low) and two additional factors: the difference between the return of the most profitable stocks and the return of the least profitable stocks (RM W , robust minus weak), and the difference between the return of stocks that invest conservatively and the return of stocks that invest aggressively (CM A, conservative minus aggressive), i.e., Ri,d − Rf,d = αi + βi (Rm,d − Rf,d ) + si SM Bd + hi HM Ld + ri RM Wd + ci CM Ad + εi,d ,

(11)

where Ri,d is the return of stock i on day d and Rf,d is the daily risk-free rate; (ii) similarly to Gatev, Goetzmann, and Rouwenhorst (2006), we use another version of daily Fama and French’s research factors including the daily Fama and French (1993)’s three factors plus momentum (Mom)12 and short-term reversal (Rev)13 factors: Ri,d − Rf,d = αi + βi (Rm,d − Rf,d ) + si SM Bd

(12)

+ hi HM Ld + mi M omd + vi Revd + εi,d . All the data used to fit the above regressions are described in and obtained from Kenneth French’s data library14 . The main purpose of these regressions is to estimate the intercept alpha - the average excess return not explained by these factors. The standard errors have been adjusted for heteroskedasticity and autocorrelation by using Newey-West adjustment with six lags. Tables 8 to 19 report the coefficients and corresponding Newey-West t-statistics of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and shortterm reversal (even numbered tables) and Fama and French (2015)’s five research factors (odd numbered tables) for each of the four strategies from 1990/2-2015, before and after transaction costs, for Top 5, Top 20, and Top 101-120 pairs. Tables 8 to 13 summarize results when positions are initiated and closed in the same day the pair diverges, and Tables 14 to 19 when we trade according “wait one day” period. For each table, Section 1 lists the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A provides results after transaction costs and Panel B before transaction costs. By analyzing Tables 8 to 11, we conclude that the copula approach usually provides larger alphas than the distance strategies, typically significant at 1%, even after accounting all the factors, for Top 5 and Top 20 pairs without delay. Thus, the results show that the variations of the profits are not fully explained by the variation of the risk factors. Tables 12 and 13 present results for Top 101-120 pairs. Here the alphas for copula method are still significantly different from zero at 1% before costs. However, the strategy does not produce significant profits on fully invested capital after costs for both multi-factors models, differently from the distance approaches. In addition, note that we usually find a positive correlation between the strategies’ average profits and the market premium when a rapid execution of the trade is made (Tables 8 to 13). The market premium slopes for distance strategies are typically larger and significant at 1%. 12 Mom is the average return on two (big and small) high prior return portfolios minus the average return on the two low prior return portfolios: M om = 1/2(Small High + Big High) − 1/2(Small Low + Big Low). 13 Rev is the average return on two (big and small) low prior return portfolios minus the average return on the two high prior return portfolios: Rev = 1/2(Small Low + Big Low) − 1/2(Small High + Big High). 14 http://http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

20

Analyzing the other risk factors from Tables 8 to 11 we can observe that the SMB coefficients are significant for copula method, at least at 10%; the value effect premium (HML factor) slopes are significant for Fama and French (2015)’s five research factors at 1% for 2.0 standard deviation threshold. We also can notice that the exposures of pairs trading return strategies to the reversal factor are positive, as suggested by Jegadeesh (1990) and Lehmann (1990), and significant at 1% for the distance strategies and usually at 1% for the copula-based pairs strategy for all regression models from Tables 8 to 19. In addition, we find that the distance strategies’ returns are negatively correlated with momentum loadings and highly significant (at 1%) as in Gatev, Goetzmann, and Rouwenhorst (2006) for all regressions. Furthermore, the return spread of firms that invest conservatively minus aggressively (CMA factor) is negatively correlated with 2.0 standard deviation strategy returns at least at 10% for regression in Tables 9 to 11. From Tables 12 and 13 one could also observe that: (1) the SMB factor is negatively correlated with profits generated by distance methods at 1% for Fama and French (1993)’s three research factors plus momentum and short-term reversal and at least at 10% for 2.0σ threshold and Fama and French (2015)’s five research factors; (2) the HML factor is positively correlated with the excess returns for 2.0σ trigger point at 10% for Fama and French (2015)’s five research factors. When we delay trades by one day (Tables 14 to 19), one could observe that the intercepts for copula strategy are significant before costs, usually at least at 5% for the Fama and French (2015)’s five research factors, whereas the distance approaches typically produce economically larger and significant alphas after costs, at least at 10%. In addition, we can note that the market premium coefficients are not significant for the copula-based strategy. The SMB factor is negatively correlated with the excess returns from copula method at 10% for Top 5 pairs and at least at 5% with the profits generated by 2.0 standard deviation threshold. Moreover, the HML factor is positively associated with the returns for the 2.0 standard deviation trigger point at least at 5% for Top 5 and Top 20 pairs when regressing the excess returns onto the Fama and French (2015)’s five research factors. Finally, the profits from the 2.0σ strategy is negatively correlated with the CMA factor at 5% for Top 5 and Top 20 pairs. The other risk-factors do not explain significantly the excess returns of the pairs trading strategies. It should be noted that the results show that the exposures to the various sources of systematic profile risk provide a low explanation of the variations of the average excess returns for any strategy as measured by the R-square and adjusted R-squared, particularly for the copula-based pairs strategy, indicating that the method is nearly factor-neutral over the whole sample period.

21

22

0.0265 0.0170 0.0602

Distance (2.0σ) Distance (0.75σ) Copula-GARCH

0.0567 0.0431 0.0426

0.0566 0.0431 0.0434

2.2840∗∗ 1.4741 2.5625∗∗

2.4462∗∗ 1.7780∗ 4.6260∗∗∗

1.8925 2.3594∗∗ 4.7154∗∗∗

0.0359 0.0399 0.0235

0.0359 0.0400 0.0233

1.7130∗ 2.0385∗∗ 2.5461∗∗

∗

Rm-Rf

t-stat

3.6615∗∗∗ 3.1453∗∗∗ 2.5406∗∗

3.6663∗∗∗ 3.1493∗∗∗ 2.5243∗∗

3.6313 3.7301∗∗∗ 3.0533∗∗∗

∗∗∗

3.6379∗∗∗ 3.7355∗∗∗ 3.0429∗∗∗

t-stat

t-stat

HML

t-stat

-1.6003 -0.7210 −1.8464∗

0.0322 0.0270 0.0179

1.5886 1.2780 1.2359

-1.6310 -0.7431 −1.8366∗

0.0325 0.0276 0.0186

1.6044 1.3020 1.2779

−1.9138∗ -0.7187 -1.2551 0.0529 0.0323 0.0508

1.5816 1.1664 1.3539

-0.0472 -0.0166 -0.0361

−1.9286∗ -0.7304 -1.2512 0.0535 0.0329 0.0529

1.5950 1.1867 1.3953

Panel B: Before Transaction Costs

-0.0468 -0.0164 -0.0358

Section 2: Return on Fully Invested Capital Panel A: After Transaction Costs

-0.0262 -0.0133 -0.0202


-0.0258 -0.0129 -0.0201

Section 1: Return on Committed Capital Panel A: After Transaction Costs

SMB

-0.0744 -0.0589 0.0144

-0.0744 -0.0588 0.0130

-0.0418 -0.0478 0.0024

-0.0417 -0.0478 0.0020

Mom

−4.2836∗∗∗ −3.6760∗∗∗ 0.5881

−4.2842∗∗∗ −3.6745∗∗∗ 0.5355

−4.1697∗∗∗ −3.8942∗∗∗ 0.2600

−4.1624∗∗∗ −3.8903∗∗∗ 0.2223

t-stat

0.0952 0.0828 0.0571

0.0946 0.0823 0.0559

0.0677 0.0670 0.0196

0.0673 0.0666 0.0189

Rev

4.6979∗∗∗ 4.4624∗∗∗ 2.0295∗∗

4.6658∗∗∗ 4.4385∗∗∗ 2.0049∗∗

4.8956∗∗∗ 4.2476∗∗∗ 1.8493∗

4.8592∗∗∗ 4.2153∗∗∗ 1.7996∗

t-stat

0.0278 0.0238 0.0059

0.0277 0.0238 0.0059

0.0369 0.0325 0.0089

0.0366 0.0324 0.0089

R2

0.0270 0.0230 0.0051

0.0269 0.0230 0.0051

0.0361 0.0318 0.0080

0.0359 0.0317 0.0081

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and short-term reversal over July 1991 and December 2015 (6173 observations) for pairs traded in the same day the pair diverges. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

0.0246 0.0141 0.0328

0.0116 0.0159 0.0226



0.0105 0.0137 0.0121

Intercept

Systematic risk of Top 5 pairs without delay: Fama and French (1993)’s three factors plus Momentum and Short-Term Reversal.


Strategy

Table 8:

23

0.0316 0.0210 0.0661


0.0879 0.0742 0.0477

0.0880 0.0744 0.0486

2.7715∗∗∗ 1.9072∗ 3.0671∗∗∗

2.9380∗∗∗ 2.2155∗∗ 5.1680∗∗∗

2.6517 2.9677∗∗∗ 5.2925∗∗∗

0.0540 0.0614 0.0240

0.0539 0.0613 0.0236

2.4664∗∗ 2.6401∗∗∗ 3.0649∗∗∗

∗∗∗

Rm-Rf

t-stat

5.4738∗∗∗ 5.0509∗∗∗ 2.7251∗∗∗

5.4660∗∗∗ 5.0434∗∗∗ 2.7034∗∗∗

4.9290 5.0108∗∗∗ 3.1965∗∗∗

∗∗∗

4.9199∗∗∗ 5.0021∗∗∗ 3.1722∗∗∗

t-stat

t-stat

HML

t-stat

-1.4376 -0.6882 −1.7680∗

0.0653 0.0570 0.0223

2.7082∗∗∗ 2.0740∗∗ 1.2661

-1.4637 -0.7107 −1.7602∗

0.0655 0.0573 0.0227

2.7185∗∗∗ 2.0845∗∗ 1.2737

-1.5056 -0.4638 -1.2600

0.1066 0.0614 0.0473

3.1742∗∗∗ 1.8534∗ 1.0261

-0.0446 -0.0124 -0.0400

-1.5174 -0.4771 -1.2473

0.1070 0.0618 0.0481

3.1842∗∗∗ 1.8626∗ 1.0336


-0.0443 -0.0120 -0.0398


-0.0281 -0.0146 -0.0230


-0.0276 -0.0141 -0.0229


SMB

0.0139 0.0217 -0.0259

0.0140 0.0220 -0.0263

-0.0072 -0.0024 -0.0148

-0.0071 -0.0021 -0.0147

RMW

0.2890 0.6033 -0.5493

0.2914 0.6128 -0.5657

-0.2707 -0.0890 -0.7719

-0.2653 -0.0788 -0.7745

t-stat

-0.0926 -0.0407 -0.0245

-0.0927 -0.0413 -0.0257

-0.0631 -0.0467 -0.0197

-0.0634 -0.0472 -0.0201

CMA

Systematic risk of Top 5 pairs without delay: Fama and French (2015)’s five factors.

−1.8594∗ -0.9159 -0.4407

−1.8621∗ -0.9301 -0.4672

−1.8128∗ -1.2706 -0.7725

−1.8191∗ -1.2851 -0.7945

t-stat

0.0168 0.0127 0.0041

0.0168 0.0127 0.0041

0.0218 0.0188 0.0078

0.0217 0.0188 0.0079

R2

0.0160 0.0119 0.0033

0.0160 0.0119 0.0033

0.0210 0.0180 0.0070

0.0209 0.0180 0.0071

2 Radj

Note: This table shows results of regressing daily return series onto Fama and French (2015)’s five research factors over July 1991 and December 2015 (6173 observations) for pairs traded and closed in the same day the pair diverges. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗∗∗ ∗∗ ∗ , , significant at 1%, 5% and 10% levels, respectively.

0.0297 0.0180 0.0386

0.0160 0.0198 0.0248



0.0148 0.0175 0.0142

Intercept


Strategy

Table 9:

24

0.0183 0.0174 0.0621


0.0311 0.0206 0.0200

0.0311 0.0206 0.0210

2.2657∗∗ 2.4635∗∗ 3.2494∗∗∗

2.5023∗∗ 2.9392∗∗∗ 6.6118∗∗∗

2.4071 3.5326∗∗∗ 7.7275∗∗∗

0.0148 0.0145 0.0080

0.0148 0.0145 0.0077

2.1394∗∗ 3.0451∗∗∗ 4.4100∗∗∗

∗∗

Rm-Rf

t-stat

3.2084∗∗∗ 2.3362∗∗ 1.7923∗

3.2074∗∗∗ 2.3393∗∗ 1.7209∗

2.7897 2.3679∗∗ 2.0608∗∗

∗∗∗

2.7986∗∗∗ 2.3771∗∗ 2.0105∗∗

t-stat

t-stat

HML

t-stat

-0.5732 -0.5300 −2.1659∗∗

0.0119 0.0149 0.0025

0.8865 1.0594 0.3048

-0.6122 -0.5674 −2.1669∗∗

0.0123 0.0155 0.0031

0.9160 1.0952 0.3795

-0.3700 -0.7714 −2.0722∗∗

0.0475 0.0365 0.0062

2.0306∗∗ 1.8910∗ 0.2224

-0.0080 -0.0148 -0.0652

-0.3934 -0.8009 −2.0670∗∗

0.0481 0.0372 0.0078

2.0540∗∗ 1.9219 0.2777


-0.0076 -0.0142 -0.0652


-0.0064 -0.0066 -0.0127


-0.0060 -0.0062 -0.0126


SMB

-0.0481 -0.0394 0.0200

-0.0480 -0.0393 0.0190

-0.0248 -0.0272 0.0040

-0.0248 -0.0271 0.0037

Mom

−2.9779∗∗∗ −3.0161∗∗∗ 1.2076

−2.9748∗∗∗ −3.0043∗∗∗ 1.1606

−3.2115∗∗∗ −3.0789∗∗∗ 0.9299

−3.2072∗∗∗ −3.0683∗∗∗ 0.8750

t-stat

0.0741 0.0649 0.0474

0.0734 0.0643 0.0465

0.0423 0.0409 0.0173

0.0418 0.0404 0.0167

Rev

4.8838∗∗∗ 4.6167∗∗∗ 2.7118∗∗∗

4.8436∗∗∗ 4.5762∗∗∗ 2.6897∗∗∗

4.7311∗∗∗ 4.2143∗∗∗ 3.1689∗∗∗

4.6814∗∗∗ 4.1616∗∗∗ 3.1007∗∗∗

t-stat

0.0323 0.0308 0.0072

0.0320 0.0305 0.0071

0.0300 0.0248 0.0082

0.0296 0.0244 0.0080

R2

0.0315 0.0300 0.0064

0.0312 0.0297 0.0063

0.0292 0.0240 0.0074

0.0288 0.0237 0.0072

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and short-term reversal from July 1991 to December 2015 (6173 observations) for pairs traded in the same day the pair diverges. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0166 0.0146 0.0301

0.0093 0.0150 0.0227



0.0082 0.0129 0.0128

Intercept

Systematic risk of Top 20 pairs without delay: Fama and French (1993)’s three factors plus Momentum and Short-Term Reversal.


Strategy

Table 10:

25

0.0230 0.0209 0.0668


0.0518 0.0426 0.0268

0.0520 0.0428 0.0280

3.2356∗∗∗ 3.5917∗∗∗ 7.2601∗∗∗

0.0267 0.0294 0.0119

0.0266 0.0292 0.0114

Rm-Rf

2.9873∗∗∗ 3.1047∗∗∗ 3.8296∗∗∗

3.0949 4.0260∗∗∗ 8.5316∗∗∗

∗∗∗

2.8204∗∗∗ 3.5317∗∗∗ 5.0810∗∗∗

t-stat

5.0534∗∗∗ 4.4207∗∗∗ 2.3198∗∗

5.0304∗∗∗ 4.4030∗∗∗ 2.2449∗∗

4.7980 4.5528∗∗∗ 2.9749∗∗∗

∗∗∗

4.7757∗∗∗ 4.5334∗∗∗ 2.8961∗∗∗

t-stat

t-stat

HML

t-stat

-0.4478 -0.1514 −1.7413∗

0.0313 0.0316 -0.0028

2.8092∗∗∗ 2.4302∗∗ -0.3446

-0.4817 -0.1840 −1.7349∗

0.0316 0.0320 -0.0026

2.8333∗∗∗ 2.4574∗∗ -0.3168

-0.3850 -0.4522 −1.8779∗

0.0842 0.0605 -0.0174

3.6304∗∗∗ 2.8994∗∗∗ -0.5456

-0.0094 -0.0099 -0.0697

-0.4058 -0.4790 −1.8723∗

0.0846 0.0611 -0.0169

3.6482∗∗∗ 2.9203∗∗∗ -0.5252


-0.0089 -0.0094 -0.0697


-0.0058 -0.0024 -0.0118


-0.0054 -0.0020 -0.0117


SMB

-0.0044 0.0203 -0.0253

-0.0044 0.0203 -0.0252

0.0022 0.0178 0.0015

0.0022 0.0178 0.0014

RMW

-0.1462 0.7977 -0.6898

-0.1454 0.8022 -0.6931

0.1272 1.0005 0.1300

0.1305 1.0036 0.1144

t-stat

-0.0678 -0.0450 0.0275

-0.0682 -0.0454 0.0261

-0.0388 -0.0304 0.0033

-0.0390 -0.0307 0.0027

CMA

Systematic risk of Top 20 pairs without delay: Fama and French (2015)’s five factors.

−1.9494∗ -1.5098 0.5974

−1.9603∗∗ -1.5242 0.5721

−1.9669∗∗ -1.4448 0.2413

−1.9798∗∗ -1.4602 0.2026

t-stat

0.0174 0.0146 0.0047

0.0173 0.0145 0.0047

0.0144 0.0118 0.0044

0.0144 0.0117 0.0043

R2

0.0166 0.0138 0.0039

0.0165 0.0137 0.0039

0.0137 0.0110 0.0036

0.0136 0.0109 0.0035

2 Radj

Note: This table shows results of regressing daily return series onto Fama and French (2015)’s five research factors from July 1991 to December 2015 (6173 observations) for pairs traded and closed in the same day the pair diverges. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0212 0.0180 0.0347

0.0120 0.0170 0.0241



0.0109 0.0149 0.0142

Intercept

Strategy Distance (2.0σ) Distance (0.75σ) Copula-GARCH

strategy

Table 11:

26

0.0308 0.0207 0.0317


0.0443 0.0288 0.0012

0.0443 0.0288 0.0018

3.4766∗∗∗ 2.6072∗∗∗ 0.2133

3.6711∗∗∗ 2.9783∗∗∗ 2.9139∗∗∗

4.2062 3.7285∗∗∗ 4.6097∗∗∗

0.0216 0.0192 0.0026

0.0217 0.0193 0.0025

3.9884∗∗∗ 3.3434∗∗∗ 1.9814∗∗

∗∗∗

Rm-Rf

t-stat

3.7255∗∗∗ 2.7643∗∗∗ 0.1127

3.7270∗∗∗ 2.7632∗∗∗ 0.0767

3.7530 2.9900∗∗∗ 0.5654

∗∗∗

3.7659∗∗∗ 3.0008∗∗∗ 0.5443

t-stat

t-stat

HML

t-stat

−3.0828∗∗∗ −3.0801∗∗∗ -0.4916 -0.0098 -0.0101 0.0029

-0.8122 -0.7101 0.3208

−3.1041∗∗∗ −3.1016∗∗∗ -0.4855 -0.0095 -0.0097 0.0034

-0.7839 -0.6804 0.3752

−2.5058∗∗ −2.3331∗∗ 0.0719 0.0114 -0.0023 0.0072

0.4654 -0.0997 0.2272

-0.0540 -0.0452 0.0029

−2.5244∗∗ −2.3516∗∗ 0.0933 0.0119 -0.0018 0.0090

0.4868 -0.0783 0.2818


-0.0536 -0.0447 0.0022


-0.0344 -0.0386 -0.0042


-0.0341 -0.0383 -0.0042


SMB

-0.1221 -0.0921 0.0184

-0.1219 -0.0919 0.0180

-0.0543 -0.0574 0.0064

-0.0542 -0.0573 0.0064

Mom

−8.5271∗∗∗ −6.6893∗∗∗ 0.8673

−8.5177∗∗∗ −6.6865∗∗∗ 0.8581

−7.4069∗∗∗ −6.4019∗∗∗ 1.1699

−7.3980∗∗∗ −6.3982∗∗∗ 1.1719

t-stat

0.1088 0.0988 0.0802

0.1082 0.0982 0.0790

0.0506 0.0613 0.0271

0.0502 0.0608 0.0265

Rev

6.5719∗∗∗ 6.1848∗∗∗ 2.5317∗∗

6.5385∗∗∗ 6.1567∗∗∗ 2.5163∗∗

5.6314∗∗∗ 5.7114∗∗∗ 3.2230∗∗∗

5.5927∗∗∗ 5.6721∗∗∗ 3.1875∗∗∗

t-stat

0.0677 0.0631 0.0060

0.0675 0.0629 0.0060

0.0567 0.0543 0.0085

0.0565 0.0541 0.0083

R2

0.0669 0.0623 0.0052

0.0667 0.0622 0.0052

0.0559 0.0535 0.0077

0.0557 0.0533 0.0075

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and short-term reversal from July 1991 to December 2015 (6173 observations) for pairs traded in the same day the pair diverges. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0291 0.0180 0.0023

0.0181 0.0175 0.0150



0.0171 0.0156 0.0064

Intercept

Systematic risk of Top 101-120 pairs without delay: Fama and French (1993)’s three factors plus Momentum and Short-Term Reversal.


Strategy

Table 12:

27

0.0336 0.0234 0.0380


0.0984 0.0782 0.0206

0.0985 0.0784 0.0212

3.8873∗∗∗ 3.0488∗∗∗ 0.7981

4.0887∗∗∗ 3.4291∗∗∗ 3.5308∗∗∗

4.5547 4.1135∗∗∗ 5.3758∗∗∗

0.0480 0.0511 0.0090

0.0479 0.0510 0.0088

4.3299∗∗∗ 3.7189∗∗∗ 2.6893∗∗∗

∗∗∗

Rm-Rf

t-stat

7.3146∗∗∗ 6.8149∗∗∗ 1.3641

7.3038∗∗∗ 6.7997∗∗∗ 1.3292

7.3642 7.0826∗∗∗ 1.9019∗

∗∗∗

7.3568∗∗∗ 7.0725∗∗∗ 1.8751∗

t-stat

t-stat

HML

t-stat

−2.4061∗∗ −2.2498∗∗ -0.0388 0.0109 0.0079 -0.0040

−2.4269∗∗ −2.2705∗∗ -0.0347 0.0111 0.0082 -0.0035

0.9581 0.6003 -0.3286

0.9393 0.5800 -0.3841

−1.8390∗ -1.5287 0.4218 0.0688 0.0300 -0.0143

-0.0436 -0.0322 0.0143

−1.8577∗ -1.5463 0.4406

0.0693 0.0303 -0.0126


-0.0431 -0.0318 0.0136

2.4581∗∗ 1.2871 -0.3344

2.4445∗∗ 1.2722 -0.3826


-0.0290 -0.0298 -0.0003


-0.0288 -0.0295 -0.0004


SMB

0.0588 0.0647 0.0334

0.0590 0.0647 0.0335

0.0297 0.0438 0.0113

0.0297 0.0437 0.0113

RMW

1.9438∗ 2.4648∗∗ 0.9227

1.9524∗ 2.4664∗∗ 0.9357

1.8128∗ 2.3907∗∗ 0.9903

1.8187∗ 2.3887∗∗ 1.0017

t-stat

-0.0541 -0.0206 -0.0041

-0.0544 -0.0209 -0.0035

-0.0118 -0.0074 -0.0029

-0.0121 -0.0076 -0.0025

CMA

Systematic risk of Top 101-120 pairs without delay: Fama and French (2015)’s five factors.

-1.4213 -0.6671 -0.0768

-1.4317 -0.6763 -0.0661

-0.5987 -0.3574 -0.1929

-0.6111 -0.3676 -0.1701

t-stat

0.0296 0.0251 0.0009

0.0295 0.0251 0.0009

0.0261 0.0234 0.0017

0.0261 0.0234 0.0017

R2

0.0288 0.0243 0.0000

0.0287 0.0243 0.0001

0.0253 0.0226 0.0009

0.0253 0.0226 0.0009

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (2015)’s five research factors from July 1991 to December 2015 (6173 observations) for pairs traded in the same day the pair diverges. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0318 0.0207 0.0085

0.0192 0.0190 0.0171



0.0182 0.0171 0.0085

Intercept


Strategy

Table 13:

28

1.6436 1.7118∗ 1.4339

1.4784 1.4896 0.1721

1.1550 1.9908∗∗ 1.3734

0.9736 1.6671∗ -0.7703

t-stat

0.0242 0.0019 0.0078

0.0242 0.0019 0.0076

0.0136 0.0058 0.0086

0.0136 0.0059 0.0084

Rm-Rf

1.6943∗ 0.1219 0.4618

1.6957∗ 0.1229 0.4490

1.4787 0.5832 1.1361

1.4841 0.5879 1.1051

t-stat

t-stat

HML

t-stat

-0.5179 0.4347 -1.4811

0.0220 0.0079 0.0119

1.1342 0.3844 0.8491

-0.5471 0.4137 -1.4879

0.0223 0.0084 0.0126

1.1516 0.4101 0.8931

-0.5994 0.0388 −1.8655∗

0.0324 0.0553 -0.0024

1.0041 1.1180 -0.0683

-0.0151 0.0007 -0.0545

-0.6132 0.0308 −1.8746∗

0.0329 0.0560 -0.0013

1.0197 1.1293 -0.0366


-0.0148 0.0009 -0.0540


-0.0084 0.0068 -0.0168


-0.0080 0.0072 -0.0167


SMB

-0.0674 -0.0591 -0.0044

-0.0674 -0.0592 -0.0054

-0.0389 -0.0465 0.0007

-0.0388 -0.0465 0.0003

Mom

−3.7852∗∗∗ −3.0086∗∗∗ -0.1929

−3.7852∗∗∗ −3.0177∗∗∗ -0.2364

−4.0982∗∗∗ −3.9086∗∗∗ 0.0772

−4.0880∗∗∗ −3.9023∗∗∗ 0.0347

t-stat

0.0806 0.0590 0.0653

0.0800 0.0586 0.0646

0.0563 0.0545 0.0313

0.0559 0.0540 0.0306

Rev

4.0247∗∗∗ 2.7841∗∗∗ 2.3090∗∗

3.9932∗∗∗ 2.7672∗∗∗ 2.3010∗∗

4.5691∗∗∗ 3.8647∗∗∗ 2.8809∗∗∗

4.5300∗∗∗ 3.8301∗∗∗ 2.8326∗∗∗

t-stat

0.0152 0.0073 0.0040

0.0151 0.0073 0.0040

0.0203 0.0151 0.0068

0.0201 0.0150 0.0066

R2

0.0144 0.0065 0.0032

0.0143 0.0065 0.0032

0.0195 0.0143 0.0060

0.0193 0.0142 0.0058

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and short-term reversal from July 1991 to December 2015 (6173 observations) for pairs traded according to the one-day waiting period rule. Section 1 shows the return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0176 0.0206 0.0191


0.0070 0.0135 0.0068


0.0158 0.0178 0.0023

0.0059 0.0112 -0.0038



Intercept

Systematic risk of Top 5 pairs with a one-day waiting period: Fama and French (1993)’s three factors plus Momentum and Short-Term Reversal.

Strategy

Table 14:

29

2.1154 1.9389∗ 1.9030∗

∗∗

1.9448∗ 1.6963∗ 0.6296

1.8377 2.5295∗∗ 2.0443∗∗

0.0466 0.0341 0.0176

0.0465 0.0339 0.0171

0.0264 0.0206 0.0124

0.0263 0.0205 0.0120

1.6488∗ 2.1975∗∗ -0.1693

∗

Rm-Rf

t-stat

3.2108 2.3490∗∗ 0.9902

∗∗∗

3.2011∗∗∗ 2.3411∗∗ 0.9712

2.7610 1.9461∗ 1.6260

∗∗∗

2.7513∗∗∗ 1.9370∗ 1.5803

t-stat

t-stat

HML

t-stat

-0.6606 0.1955 -1.5566

0.0581 0.0475 0.0133

2.7564∗∗∗ 1.9574∗ 0.8493

-0.6868 0.1728 -1.5591

0.0584 0.0478 0.0136

2.7684∗∗∗ 1.9699∗∗ 0.8616

-0.5476 0.6944 −1.9433∗

0.0921 0.0869 0.0004

2.9306∗∗∗ 2.0753∗∗ 0.0089

-0.0166 0.0195 -0.0637

-0.5597 0.6871 −1.9359∗

0.0925 0.0871 0.0006

2.9427∗∗∗ 2.0776∗∗ 0.0143


-0.0162 0.0197 -0.0635


-0.0127 0.0033 -0.0207


-0.0122 0.0037 -0.0206


SMB

-0.0037 0.0811 -0.0424

-0.0035 0.0811 -0.0438

-0.0170 -0.0120 -0.0194

-0.0169 -0.0117 -0.0194

RMW

-0.0781 1.2202 -0.8428

-0.0752 1.2226 -0.8763

-0.6200 -0.4418 -0.9673

-0.6150 -0.4316 -0.9698

t-stat

-0.1095 -0.0484 -0.0184

-0.1097 -0.0493 -0.0191

-0.0689 -0.0683 -0.0131

-0.0691 -0.0688 -0.0134

CMA

−2.2434∗∗ -0.8782 -0.3354

−2.2485∗∗ -0.8947 -0.3487

−2.1575∗∗ −2.0130∗∗ -0.5593

−2.1641∗∗ −2.0285∗∗ -0.5776

t-stat

Systematic risk of Top 5 pairs with a one-day waiting period: Fama and French (2015)’s five factors.

0.0075 0.0038 0.0019

0.0075 0.0038 0.0019

0.0096 0.0054 0.0032

0.0096 0.0054 0.0032

R2

0.0067 0.0030 0.0011

0.0067 0.0029 0.0011

0.0088 0.0046 0.0024

0.0088 0.0046 0.0024

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (2015)’s five research factors over July 1991 and December 2015 (6173 observations) for pairs traded according to the one-day waiting period rule. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0224 0.0215 0.0252


0.0110 0.0168 0.0098


0.0206 0.0187 0.0083

0.0098 0.0146 -0.0008



Intercept

Strategy

Table 15:

30

0.0164 0.0120 0.0236


-0.0058 -0.0192 -0.0136

-0.0058 -0.0192 -0.0133

2.3369∗∗ 1.9974∗∗ 2.2209∗∗

-0.0038 -0.0120 -0.0013

-0.0037 -0.0119 -0.0015

Rm-Rf

2.0908∗∗ 1.5588 0.3554

1.8198 2.7915∗∗∗ 3.4592∗∗∗

∗

1.5428 2.2875∗∗ 0.1005

t-stat

-0.6415 −2.2593∗∗ -1.0247

-0.6469 −2.2643∗∗ -1.0617

-0.7372 −2.0240∗∗ -0.3392

-0.7360 −2.0263∗∗ -0.4167

t-stat

t-stat

HML

t-stat

0.2061 0.7101 -0.6578

0.0020 -0.0011 0.0008

0.1641 -0.0800 0.0981

0.1677 0.6762 -0.6772

0.0024 -0.0005 0.0014

0.1983 -0.0406 0.1671

0.4988 0.4067 -1.2119

0.0122 0.0151 -0.0050

0.6036 0.7886 -0.1722

0.0095 0.0070 -0.0344

0.4750 0.3838 -1.2143

0.0128 0.0158 -0.0040

0.6342 0.8213 -0.1355


0.0100 0.0074 -0.0341


0.0017 0.0078 -0.0043


0.0021 0.0082 -0.0041


SMB

-0.0510 -0.0401 0.0228

-0.0510 -0.0399 0.0219

-0.0232 -0.0284 0.0048

-0.0231 -0.0283 0.0044

Mom

−3.8367∗∗∗ −3.3994∗∗∗ 1.4331

−3.8345∗∗∗ −3.3912∗∗∗ 1.3918

−3.2639∗∗∗ −3.4366∗∗∗ 1.1588

−3.2594∗∗∗ −3.4252∗∗∗ 1.0882

t-stat

0.0501 0.0428 0.0694

0.0494 0.0422 0.0683

0.0327 0.0324 0.0195

0.0322 0.0319 0.0189

Rev

3.6721∗∗∗ 3.1525∗∗∗ 3.8327∗∗∗

3.6289∗∗∗ 3.1154∗∗∗ 3.8087∗∗∗

4.0702∗∗∗ 3.5540∗∗∗ 3.4798∗∗∗

4.0170∗∗∗ 3.4998∗∗∗ 3.4078∗∗∗

t-stat

0.0135 0.0114 0.0053

0.0133 0.0112 0.0052

0.0136 0.0118 0.0053

0.0133 0.0115 0.0051

R2

0.0127 0.0106 0.0045

0.0125 0.0104 0.0044

0.0128 0.0110 0.0045

0.0125 0.0107 0.0042

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and short-term reversal from July 1991 to December 2015 (6173 observations) for pairs traded according to the one-day waiting period rule. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0147 0.0094 0.0037

0.0068 0.0114 0.0101



0.0057 0.0093 0.0003

Intercept

Systematic risk of Top 20 pairs with a one-day waiting period: Fama and French (1993)’s three factors plus Momentum and Short-Term Reversal.


Strategy

Table 16:

31

0.0190 0.0137 0.0301


0.0101 -0.0025 -0.0022

0.0103 -0.0023 -0.0016

2.5038∗∗ 1.8547∗ 1.0000

2.7598∗∗∗ 2.3014∗∗ 2.9442∗∗∗

2.4084 3.1947∗∗∗ 4.2145∗∗∗

0.0043 -0.0014 0.0026

0.0042 -0.0016 0.0021

2.1230∗∗ 2.6847∗∗∗ 0.7021

∗∗

Rm-Rf

t-stat

1.1155 -0.2647 -0.1164

1.0910 -0.2896 -0.1591

0.8427 -0.2450 0.6552

0.8184 -0.2724 0.5489

t-stat

t-stat

HML

t-stat

0.0976 0.8204 -0.4993

0.0240 0.0240 -0.0044

2.4015∗∗ 1.9361∗ -0.5557

0.0628 0.7891 -0.5043

0.0243 0.0244 -0.0043

2.4284∗∗ 1.9659∗∗ -0.5341

0.3533 0.6339 -1.0716

0.0523 0.0447 -0.0283

2.5516∗∗ 2.0690∗∗ -0.8546

0.0076 0.0126 -0.0346

0.3319 0.6132 -1.0636

0.0527 0.0452 -0.0278

2.5719∗∗ 2.0860∗∗ -0.8339


0.0081 0.0131 -0.0347


0.0007 0.0102 -0.0038


0.0012 0.0107 -0.0037


SMB

-0.0036 0.0252 -0.0122

-0.0036 0.0252 -0.0132

-0.0039 0.0109 -0.0008

-0.0039 0.0109 -0.0011

RMW

-0.1189 0.9554 -0.2779

-0.1176 0.9577 -0.3020

-0.2424 0.6268 -0.0636

-0.2387 0.6303 -0.0841

t-stat

-0.0635 -0.0514 0.0095

-0.0638 -0.0519 0.0098

-0.0430 -0.0473 0.0016

-0.0432 -0.0475 0.0011

CMA

−2.0552∗∗ −1.7032∗ 0.2040

−2.0675∗∗ −1.7201∗ 0.2115

−2.4068∗∗ −2.3426∗∗ 0.1087

−2.4212∗∗ −2.3591∗∗ 0.0735

t-stat

Systematic risk of Top 20 pairs with a one-day waiting period: Fama and French (2015)’s five factors.

0.0035 0.0029 0.0008

0.0034 0.0028 0.0009

0.0036 0.0026 0.0004

0.0036 0.0026 0.0004

R2

0.0026 0.0020 0.0000

0.0026 0.0020 0.0001

0.0028 0.0018 -0.0004

0.0028 0.0018 -0.0004

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (2015)’s five research factors from July 1991 to December 2015 (6173 observations) for pairs traded according to the one-day waiting period rule. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0172 0.0110 0.0101

0.0089 0.0131 0.0119



0.0079 0.0110 0.0020

Intercept


Strategy

Table 17:

32

0.0134 0.0145 0.0122

0.0198 0.0182 0.0087

0.0215 0.0207 0.0279




2.6404 2.9467∗∗∗ 2.3272∗∗

∗∗∗

2.4371∗∗ 2.5956∗∗∗ 0.7312

3.1582 3.1004∗∗∗ 3.9696∗∗∗

-0.0021 -0.0300 -0.0272

-0.0021 -0.0300 -0.0275

-0.0005 -0.0104 -0.0038

-0.0004 -0.0104 -0.0039

2.9357∗∗∗ 2.7120∗∗∗ 1.1647

∗∗∗

Rm-Rf

t-stat

-0.1851 −2.2014∗∗ -1.5079

-0.1846 −2.2044∗∗ -1.5376

-0.0878 −1.6946∗ -0.8746

-0.0776 −1.6871∗ -0.9075

t-stat

t-stat

HML

t-stat

−2.6594∗∗∗ −2.6284∗∗∗ 0.3457 -0.0211 -0.0239 -0.0078

−1.8800∗ −1.8332∗ -0.9201

−2.6849∗∗∗ −2.6542∗∗∗ 0.3486 -0.0208 -0.0235 -0.0074

−1.8462∗ −1.7969∗ -0.8552

−2.0226∗∗ -0.9706 1.3980 -0.0259 -0.0375 -0.0306

-1.1090 −1.7328∗ -0.9349

-0.0417 -0.0176 0.0474

−2.0421∗∗ -0.9915 1.4099

-0.0254 -0.0369 -0.0295

-1.0849 −1.7052∗ -0.8926


-0.0413 -0.0172 0.0467


-0.0284 -0.0312 0.0029


-0.0281 -0.0309 0.0029


SMB

-0.1098 -0.0822 0.0116

-0.1097 -0.0821 0.0116

-0.0558 -0.0587 0.0055

-0.0557 -0.0586 0.0054

Mom

−7.6438∗∗∗ −6.2318∗∗∗ 0.5584

−7.6334∗∗∗ −6.2230∗∗∗ 0.5610

−7.8249∗∗∗ −7.0596∗∗∗ 1.0975

−7.8134∗∗∗ −7.0545∗∗∗ 1.0918

t-stat

0.0790 0.0654 0.0936

0.0785 0.0649 0.0924

0.0387 0.0504 0.0305

0.0383 0.0499 0.0298

Rev

4.9589∗∗∗ 4.0119∗∗∗ 2.7261∗∗∗

4.9291∗∗∗ 3.9850∗∗∗ 2.7083∗∗∗

4.5893∗∗∗ 5.2049∗∗∗ 3.8410∗∗∗

4.5480∗∗∗ 5.1599∗∗∗ 3.7997∗∗∗

t-stat

0.0354 0.0242 0.0076

0.0352 0.0241 0.0076

0.0339 0.0326 0.0113

0.0337 0.0324 0.0111

R2

0.0346 0.0234 0.0068

0.0344 0.0233 0.0068

0.0331 0.0319 0.0105

0.0329 0.0316 0.0103

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (1993)’s three research factors plus momentum and short-term reversal from July 1991 to December 2015 (6173 observations) for pairs traded according to the one-day waiting period rule. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0124 0.0126 0.0035

Intercept

Systematic risk of Top 101-120 pairs with a one-day waiting period: Fama and French (1993)’s three factors plus Momentum and Short-Term


Strategy

Table 18: Reversal.

33

0.0222 0.0216 0.0360


0.0441 0.0083 -0.0122

0.0442 0.0085 -0.0117

2.5948∗∗∗ 2.7718∗∗∗ 1.4054

2.8078∗∗∗ 3.1346∗∗∗ 3.0136∗∗∗

3.2926 3.3537∗∗∗ 4.7639∗∗∗

0.0226 0.0170 0.0026

0.0225 0.0169 0.0024

3.0608∗∗∗ 2.9542∗∗∗ 1.9538∗

∗∗∗

Rm-Rf

t-stat

3.5044∗∗∗ 0.5509 -0.6260

3.4928∗∗∗ 0.5366 -0.6551

3.7662 2.5981∗∗∗ 0.5799

∗∗∗

3.7573∗∗∗ 2.5822∗∗∗ 0.5389

t-stat

t-stat

HML

t-stat

−2.1129∗∗ −2.0558∗∗ 0.9147 0.0032 -0.0010 -0.0098

−2.1378∗∗ −2.0807∗∗ 0.9165 0.0034 -0.0007 -0.0093

0.3162 -0.0555 -0.9262

0.2951 -0.0777 -0.9848

-1.3662 -0.6281 1.7106∗

0.0248 -0.0120 -0.0208

-0.0305 -0.0118 0.0597

-1.3866 -0.6497 1.7204∗

0.0252 -0.0116 -0.0193


-0.0300 -0.0114 0.0590

0.9968 -0.4625 -0.4860

0.9820 -0.4778 -0.5287


-0.0242 -0.0256 0.0081


-0.0239 -0.0253 0.0080


SMB

0.0629 0.0381 0.0318

0.0631 0.0382 0.0319

0.0263 0.0318 0.0152

0.0264 0.0317 0.0152

RMW

2.1700∗∗ 1.4701 0.7660

2.1787∗∗ 1.4724 0.7742

1.6422 1.7642∗ 1.3533

1.6477∗ 1.7620∗ 1.3619

t-stat

-0.0399 0.0042 -0.0891

-0.0404 0.0038 -0.0874

-0.0144 -0.0125 -0.0175

-0.0146 -0.0127 -0.0171

CMA

-1.1453 0.1259 -1.5160

-1.1586 0.1143 -1.4980

-0.7691 -0.6208 -1.1488

-0.7831 -0.6328 -1.1294

t-stat

Systematic risk of Top 101-120 pairs with a one-day waiting period: Fama and French (2015)’s five factors.

0.0079 0.0012 0.0028

0.0079 0.0012 0.0029

0.0081 0.0054 0.0027

0.0081 0.0054 0.0027

R2

0.0071 0.0004 0.0020

0.0071 0.0004 0.0020

0.0073 0.0046 0.0019

0.0073 0.0046 0.0019

2 Radj

Note: This table shows results of regressing daily portfolio return series onto Fama and French (2015)’s five research factors from July 1991 to December 2015 (6173 observations) for pairs traded according to the one-day waiting period rule. Section 1 shows the Return on Committed Capital and Section 2 on Fully Invested Capital. Panel A lists the results after transaction costs and Panel B before transaction costs. The t-statistics are computed using Newey-West standard errors with six lags. ∗ ∗∗ ∗∗∗ . . significant at 1%. 5% and 10% levels, respectively.

0.0205 0.0190 0.0167

0.0137 0.0154 0.0147



0.0127 0.0135 0.0060

Intercept


Strategy

Table 19:

3.5

Robustness checks of the performance of Excess Returns and Sharpe Ratios

One possible criticism might be that the conclusions are based on only one realization of the stochastic process of asset returns computed from the observed series of prices, since among thousands of different strategies is very likely that we find some that show superior performance in terms of excess return or Sharpe Ratio. In order to mitigate data-snooping criticisms, we use the stationary bootstrap15 of Politis and Romano (1994) to compute the bootstrapped p-values using the methodology proposed by Ledoit and Wolf (2008). The bootstrap method is used to obtain the distribution of a null hypothesis. Here we want to investigate if the average excess return and the Sharpe Ratio of the Copula methods beat the reference (distance) strategies. To construct the distributions, we bootstrapped the original time series B=10,000 times. Our bootstrapped null distributions are derived from Theorem 2 of Politis and Romano (1994). We select the optimal block length for the stationary bootstrap following Politis and White (2004). As the optimal bootstrap block-length is different for each strategy, we average16 the block-lengths found to proceed the comparisons between the Copula and the benchmark strategies. To test the hypotheses that the average excess returns and Sharpe Ratios of copula strategy are equal to that of distance methods, that is, H0 : µc − µd = 0 and H0 :

µc µd − = 0, σc σd

(13)

we compute, following Davison and Hinkley (1997), a two-sized p-value using B = 10, 000 (stationary) bootstrap re-samples as follows:

psboot

 PB ∗(b) 2 b=1 I{0 0, B+1 PB = I{0≥t∗(b) }+1  2 b=1 B+1 , otherwise,

(14)

where I is the indicator function, t∗(b) are the values in each block stationary bootstrap replication, and B denotes the number of bootstrap replications. Table 20 reports the bootstrapped p-values for testing the null hypotheses represented by (13). We compare the copula method with each of the distance approaches (0.75 and 2.0 standard deviations) from 1991/2-2015 for all scenarios, i.e., without delay and waiting one-day period for the Top 5, Top 20, and Top 101-120 pairs, respectively, and before and after costs. Overall, these results reinforce the ones previously obtained. As can be observed, the copula approach significantly outperforms the distance strategies when ignoring the costs and when a rapid execution of the trade is made for Top 5 and Top 20 pairs, in particular in terms of risk-adjusted returns for both weighting structures. 15 It

allows for weakly dependent correlation over time. also use the optimal block size for each strategy. We find that the results are robust to the optimal block size, and therefore, we do not report them here. 16 We

34

Table 20: Bootstrap p-values computed from B=10,000 replications for testing the null hypotheses of equality of the average excess returns and Sharpe Ratios over the period between July 1991 and December 2015. Copula vs Distance (0.75σ) Scenario

Return

Sharpe Ratio

Copula vs Distance (2.0σ) Return

Sharpe Ratio

0.2490 0.7962 0.8464 0.1468 0.0080∗∗∗ (>) 0.5154 0.4792 0.1848 0.5380 0.0464∗∗ () 0.6960 0.8692 0.1844 0.0000∗∗∗ (>) 0.0752∗ (>) 0.1384 0.3062 0.6622 0.2112 0.3122 0.4148

0.0462∗∗ (>) 0.6622 0.9602 0.3564 0.0004∗∗∗ (>) 0.2828 0.3970 0.5218 0.8958 0.0452∗∗ () 0.8478 0.8672 0.2894 0.0086∗∗∗ (>) 0.6538 0.9732 0.2406 0.5714 0.0148∗∗ () 0.1580 0.1932 0.8636 0.3106 0.2292 0.3380 0.7880

0.0322∗∗ (>) 0.4498 0.9472 0.3498 0.0080∗∗∗ (>) 0.7352 0.7160 0.4710 0.8752 0.0544∗∗∗ () and () 0.1244 0.7814 0.5350 0.0008∗∗∗ (>) 0.0410∗∗ (>) 0.1308 0.8390 0.9880 0.4166 0.5912 0.7846

0.0016∗∗∗ (>) 0.0224∗∗ (>) 0.5614 0.6320 0.0000∗∗∗ (>) 0.0004∗∗∗ (>) 0.0138∗∗ (>) 0.4142 0.3018 0.9380 0.0912∗ (>) 0.5520

0.0004∗∗∗ (>) 0.0176∗∗ (>) 0.8346 0.7062 0.0000∗∗∗ (>) 0.0010∗∗∗ (>) 0.0890∗ (>) 0.4640 0.1272 0.6910 0.0470∗∗ (>) 0.1924

0.0036∗∗∗ (>) 0.0412∗∗ (>) 0.9774 0.5830 0.0000∗∗∗ (>) 0.0042∗∗∗ (>) 0.3692 0.8886 0.4340 0.8386 0.2574 0.5740

Section 1: Committed Capital d0p5c0 d0p5c1 d1p5c0 d1p5c1 d0p20c0 d0p20c1 d1p20c0 d1p20c1 d0p101c0 d0p101c1 d1p101c0 d1p101c1

0.2862 0.7096 0.5002 0.1574 0.1134 0.5824 0.7586 0.4178 0.7660 0.3328 0.8996 0.5790

0.0174∗∗ (>) 0.1466 0.9110 0.2648 0.0000∗∗∗ (>) 0.0136∗∗ (>) 0.0830∗ (>) 0.9110 0.3432 0.9242 0.1396 0.6530

d0p5c0 d0p5c1 d1p5c0 d1p5c1 d0p20c0 d0p20c1 d1p20c0 d1p20c1 d0p101c0 d0p101c1 d1p101c0 d1p101c1

0.0030∗∗∗ (>) 0.0430∗∗ (>) 0.8012 0.7674 0.0000∗∗∗ (>) 0.0014∗∗∗ (>) 0.0324∗∗ (>) 0.2262 0.0434∗∗ (>) 0.4176 0.0378∗∗ (>) 0.1708

0.0400∗∗ (>) 0.2024 0.9528 0.6314 0.0000∗∗∗ (>) 0.0230∗∗ (>) 0.2132 0.5686 0.5282 0.7758 0.4352 0.7622

Section 2: Fully Invested Capital

Note: This table reports the bootstrap p-values for testing the null hypothesis of equality of the average excess returns and the Sharpe Ratios of Copula and distance strategies for the Top 5, 20 and 101-120 pairs over the period between July 1991 and December 1999 (2149 observations). The column labeled Scenario contains symbol labels for trading with no delay or one day waiting period (d0 and d1, respectively), for Top 5, 20 and 101-120 pairs (p5, p20 and p101, respectively) and before or after costs (c0 and c1, respectively). The symbol labels (>) and (